A4 Paper Stories

Summary

A4 Paper Stories By Susam Pal on 06 Jan 2026 I have a measuring technique that is neither fast, nor accurate, nor recommended by any standards body and yet it hasn't failed me whenever I have had to use it. I'll share it here but don't ever use it for installing kitchen cabinets or anything that will stare back at you every day for the next ten years. It involves one tool: a sheet of A4 paper. Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go. Nevertheless, I often myself needing to measure something at short notice, usually in situations where a certain amount of inaccuracy is entirely forgivable. When I cannot fetch a ruler easily, I rely on the next best thing: a sheet of A4 paper, which is available in abundant supply where I live. From photocopying night-sky charts to serving as a scratch pad for working through mathematical proofs, A4 paper has been a trusted companion since my childhood days. I use it often. If I am carrying a bag, there is almost always some A4 paper inside: perhaps a printed research paper or a mathematical problem I have worked on recently and need to chew on a bit more during my next train ride. Dimensions The dimensions of A4 paper are the solution to a simple, elegant problem. Imagine designing a sheet of paper such that, when you cut it in half parallel to its shorter side, both halves have exactly the same aspect ratio as the original. In other words, if the shorter side has length \( x \) and the longer side has length \( y , \) then \[ \frac{y}{x} = \frac{x}{y / 2} \] which gives us \[ \frac{y}{x} = \sqrt{2}. \] Test it out. Suppose we have \( y/x = \sqrt{2}. \) We cut the paper in half parallel to the shorter side to get two halves, each with shorter side \( x' = y / 2 = x \sqrt{2} / 2 = x / \sqrt{2} \) and longer side \( y' = x. \) Then indeed \[ \frac{y'}{x'} = \frac{x}{x / \sqrt{2}} = \sqrt{2}. \] In fact, we can keep cutting the halves like this and we'll keep getting ...